EnPassant

My argument is to define X without X as a subset of itself regardless of whether it can or can't be such. In this way the paradox is avoided by defini...

My idea is that it can be framed in terms of set theory alone without the invention of classes. My definition is that if X as a subset of itself is ex...

This still begs the question about the difference between a limit and an actual infinite sum. Your reasoning shows that you don't run out of natural n...

The reason the unit circle is a standard in geometry is because what applies to the unit applies to a circle of radius 10 or 100 or 1000. It is the in...

But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit width.

But you still have 0 + 0 + ... = something

Say 0 + 0 + 0 + ... = 50 units. Simplify- (0 + 0 + 0 + ...)/50 = 1 But 0/50 = 0. It's just a matter of reducing to the lowest terms and the same logic...

That was a typo, I meant to say 9/10 + 9/100 + ... Yes, because it can't be defined in terms of calculus but the question remains, what is it? Yes, bu...

It's the same idea 0.999.. = 9(1/10 + 1/100 + ...) What seems to be happening here is that 1/x = 0 at infinity. Then you have the absurd(???) conclusi...

Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? If we're talking about an infinite sum the same applies: by that way of looking at...

Ok, but can't this be also said for 0.999...? Adding terms and then saying 'at infinity'. You can't have (b) at any finite number of the terms 9/10, 9...

Yes, you got it. The point I'm making is that 1/k is positive and > 0. Even as you go to infinity 1/x can't be zero. So, you are summing an infinity o...

I'm not talking about the 1024th term. I'm saying- 1/2+1/4+1/8+...+ 1/2^c to k terms - whatever the value of k >1/2^c + 1/2^c + 1/2^c to k terms. Ther...

Personally I think mathematics is not really about numbers. Mathematics is more about harmonies and proportion. Numbers are 'markers' in the symphony ...

I'm not up to speed on binary. I don't think you understand what I'm saying. Are all of the terms in 1/2, 1/4, 1/8...positive and > 0? Yes. For all c ...

1/2 + 1/4 + 1/8+....+1/1024 > 1/1024 + 1/1024 + 1/1024+....+1/1024 LHS has the same number of terms as RHS Now let the number of terms run to infinity...

It doesn't matter. An infinite sum of equal infinitesimals must be infinite.

What is 1/2^c added to itself infinitely? 1/2^c added 2^c times is 1 Now add another 2^c terms = 1 1 + 1 + ... = \infty

The sum of terms 1/x is infinity if 1/x > 0.

x is a power of 2. 1/2 + 1/4 + 1/8+....+1/1024 > 1/1024 + 1/1024 + 1/1024+....+1/1024 for the same number of terms. Now go to infinity with x. You are...

x is not static. I'm saying if there are the same number of terms in each. Now increase x indefinitely with the same number of terms top and bottom.

1/2 + 1/4 + 1/8+....1/x > 1/x + 1/x + 1/x+...+1/x Now let x go to infinity- \sum_{i=1}^{\infty} 1/x = \infty

I don't insist it doesn't sum to 1. You may well be right. I'm saying we don't know because we can never have an actual infinite sum. An infinity of p...

But that begs the question: you say it don't sum to infinity because it don't sum to infinity. That is the very thing that is being questioned. I know...

I know, I made a mistake. Let me rethink how to formulate it...

rethinking...

What I'm saying is very simple. Suppose you had a kind of God calculator that would print out the actual addition of 9/10 + 9/100... to an infinity of...

I missed this post. Yes, the definition of the sum is the same as the limit. But I am talking about an actual sum. An explicit infinite sum that you c...

No. You have to account for social factors. Only in affluent societies do women have the means to peruse science etc. Women don't go into scientific a...

1/9 is a proportional relationship. In geometry if the side of the square is 1 the diagonal is \sqrt{2} and the proportion is 1:\sqrt{2}

No, we know what a limit is. But as I keep saying, a limit is not a literal infinite sum. Weierstrass did not formulate calculus in terms of literal i...

https://en.wikibooks.org/wiki/Calculus/Formal_Definition_of_the_Limit

I'm aware of that. But it has not been explicitly defined. It has been defined in terms of limits which are limits of finite sums.

They compute limits which are not the same as sums. A limit is what a finite sum converges to.

See next answer. That still begs the question what is an infinite sum if nobody has ever computed it? You can't jump from the finite to the infinite a...

Emphasis mine.

It hardly matters. An infinite sum is undefined because nobody has ever computed it. The problem is that while it is ok to apply the logic to finite q...

They don't say the infinite sum, ie the sum of all terms.

You need to provide the link. As a working definition that may well be useful but there are still problems with infinite sums.

By a finite number of terms. Yes, you are correct but what is an infinite sum of terms? What happens when you sum an infinity of terms? Calculus does ...

No. Calculus is formulated in terms of finite sums and limits. You can't jump to infinity and expect the rules of finite arithmetic to apply. Jumping ...

You need to read back a few pages to see what I'm saying. It is like this- It is being asserted that 1/10 + 1/100 + ...taken to an infinity of terms i...

Earlier on someone wrote a very convincing 'proof': x = 0.999... 10x = 9.999... 10x = 9 + 0.999... 10x = 9 + x 9x = 9 x = 1 All well and good. But wha...

But this is what is being asserted: 1/10 + 1/100 + ...taken to an infinite sum of terms. Let S be this literal infinite sum. What is S? Is it 1 or \in...

In geometry the length of the line - in this example \sqrt{2} - is exact. But the decimal expansion representing it is not, unless we go to an infinit...

It may well be that the infinite sum is 1 but mathematicians were suspicious about such a concept because infinity is not a number. This is why calcul...

Yes exactly, calculus works in practice. You can sum 10^{1000000000000000000000000} terms and that's fine because it is a finite sum. But what is 10^{...

Kummer did not believe in real numbers; "God made the integers and all the rest is the work of man". This is a bit extreme as real numbers - whatever ...

I understand what you are saying but a literal infinite sum is not considered in calculus. If partial sums are added they approach the limit. If more ...

Yes, 'partial' sums. That means the sums are finite. Calculus does not speak about literal infinite sums. It speaks about finite sums approaching a li...